Remark. Every field is contained in some Pythagorean field. The smallest Pythagorean field over a field F is called the Pythagorean closure of F, and is written Fp⁢y. Given a field F, one way to construct its Pythagorean closure is as follows: let K be an extension over F such that there is a tower

F=K1⊆K2⊆⋯⊆Kn=K

of fields with Ki+1=Ki⁢(1+αi2) for some αi∈Ki, where i=1,…,n-1. Take the compositumL of the family 𝒦 of all such K’s. Then L=Fp⁢y.